A Theory of the Risk for Empirical CVaR with Application to Portfolio Selection

When decisions are based on empirical observations, a trade-off arises between flexibility of the decision and ability to generalize to new situations. In this paper, we focus on decisions that are obtained by the empirical minimization of the Conditional Value-at-Risk (CVaR) and argue that in CVaR the trade-off between flexibility and generalization can be understood on the ground of theoretical results under very general assumptions on the system that generates the observations. The results have implications on topics related to order and structure selection in various applications where the CVaR risk-measure is used. A study on a portfolio optimization problem with real data demonstrates our results.